This is not the kind of trust you have in any particular person or situation, because you feel confident that things will turn out okay. It's about knowing that you'll be okay even if things *don't* turn out well, even if (at the most extreme) you were to die. To the degree I have basic trust, I also have equanimity. Basic trust may even be yet another term for the need behind all the needs in Nonviolent Communication.
I got this term from the sixth chapter of A. H. Almaas' Facets of Unity. I knew what he was talking about within a page or so, and kept getting almost bored as he went on about it, and each time would be suddenly deeply engaged again by his further articulation of what basic trust is, and how its presence — or lack thereof — manifests.
I wrote this "root" (something like a spell) to support my basic trust at the Story Field Conference in 2007:
I am held
I am loved
Every breath my love enters my lungs
She kisses them from the inside
Every burned branch
Tenderly caring for my being
Awakening in my soul, in every cell,
And i know that i am a part
(Even though i may seem apart)
of concentrically messy
that reach out their arms to embrace all
S/he talks with me
whispers sweet nothings inside of me
If i am quiet i can hear her
Converse with her
And she always loves to hear what i have to say.